1. Introduction

The equation for heterodyne optical power P {W} (i.e., equivalent optical power generating the heterodyne receiver signal) defines the performance of the coherent lidar in the effective solid angle Ω_COH {sr} at range R {m}

where the calibration constant C {W m} groups the conversion efficiencies and parameters that describe the various system components, β is the aerosol volume backscatter coefficient {m^-1 sr ^-1 }, and α is the atmospheric linear extinction coefficient {m^-1 }. The coherent solid angle Ω_COH or coherent responsivity is expressed in terms of the maximum available solid angle A_R /R² used in incoherent lidars and the system-antenna efficiency η_S (R,t) that describes the extent to which this value is degraded [1, 2]. Here, A_R is the aperture {m² } of the lidar telescope.

At optical wavelengths, the effects of refractive turbulence on ground-based systems are certainly among the most important factors that degrade the system-antenna efficiency and, consequently, decrease the coherent signal power of a lidar. Theoretical research on the problem accomplishes calculations through considerations of the higher moments of the field and is still difficult. By using numerical simulations, we overcome the analytical limitations and are able to study the effects of atmospheric refractive turbulence on coherent lidar performance in a realistic way [3]. By use of the two-beam model [4], the approach expresses the effective solid angle in terms of the overlap integral of the transmitter and the virtual (backpropagated) local oscillator beams at the target, reducing the problem to one of computing irradiance along the two propagation paths. Recently, proceeding from numerical simulations, some analytical empirical expressions were derived for the heterodyne intensity for both monostatic and bistatic lidars [5].

The standard technique, in which the atmosphere is modeled as a set of two-dimensional Gaussian random phase screens [6, 7], is based on the Fresnel approximation to the wave equation and has provided the tools for analyzing heterodyne lidars with general turbulence conditions and arbitrary receiver configurations. Addressing all significant mechanisms on the performance of the coherent lidar, we considered the dependencies of refractive turbulence effects on near field [8] and reflected on the problem of angular misalignment caused by the presence of turbulence [9]. Also, we pondered the uncertainty caused by the presence of turbulence that is inherent to the process of optical power measurement with a practical coherent lidar [10]. The concept of beam averaging was essential to interpreting the results of our analysis (the coherent return is evaluated in the target plane as the overlap integral of the transmitted and virtual local oscillator beams, so when a large number of bright spot scintillations are found in the target plane, we should expect an averaging principle to apply). This latter work preludes use of the simulation technique to take account of the effects of refractive turbulence in the performance of a heterodyne differential absorption lidar (DIAL) system, where the error associated with the measurement depends on the estimation error of the power received signal.

Coherent lidars based on 2-µm wavelength lasers, such as those built upon diode-pumped Ho:Tm:YLF crystals, have been used extensively for wind profiling. Because of the rich spectrum of absorption features with the tuning range of these lasers, their capability for making DIAL measurements of both water vapor (H2O) and carbon dioxide (CO₂) has been realized [11, 12]. Reliable observations of these species are critical to many atmospheric studies. Coherent DIAL results have also been reported with the use of CO₂ lasers at the 10-µm wavelength region. The maturity of CO₂ technology for heterodyne detection and the existence at this wavelength region of multiple absorption lines of H2O [13] and pollutants such as ammonia (NH3) [14]—one of the most important originators of aerosol formation, a major pollutant in urban environments—have excited the development of instruments that combine wind and DIAL measurement capabilities [15].

Results will be shown for both 2- and 10-µm coherent DIAL systems and those previously mentioned molecular species. Choosing these target species, although they are most significant in today’s atmospheric and climate research programs, is a matter of practicality: Results shown in our figures and any conclusion extracted from them are fully generalizable to any coherent lidar regardless of the molecular species under consideration. The primary objective of this study is to understand turbulence-induced uncertainties in coherent DIAL systems that profile the atmosphere.

2. DIAL measurement characteristics

In DIAL measurement, two lidar pulses of slightly different wavelengths λ_on and λ_off are selected such that λ _on corresponds to the absorption line of the species of interest whereas λ _off is placed in a transparent region of the species line. The radiation at the nonabsorbing wavelength λ _off provides a reference so that the scattering and absorption properties of the atmosphere can be separated from those of the absorbing gas. Selecting wavelength pairs that are as closely spaced as possible, while still providing sufficient differential absorption, will enable us to assume that differences in backscattering coefficient and extinction coefficient due to gases other than the species of interest will be negligible for the measurements at wavelengths λ _on and λ _off. As a consequence, the coherent lidar equation for the total scattered laser power at those wavelengths allows solving for the concentration of the molecule to be measured: Applying the lidar Eq. (1) to both the on-line and off-line backscatter signals, taking the logarithm of the ratio, and estimating the derivative, we finish with the DIAL equation [16]

with δP being the ratio of the received powers from the on-line beam P _on and the off-line beam P _off

Here, ΔR {m} is the range resolution, ρ is the mean concentration measurement {molecule/m³ } between ranges R-ΔR/2 and R+ΔR/2, and the differential absorption coefficient K is the remainder σ _on -σ _off , where σ=α/ρ is the molecular cross section {m ²/molecule}. From Eq. (1) we can extract that Eq. (3) shows no dependency with the calibration constant C, so the DIAL measurement relies only on the coherent solid angle Ω_COH . Equations (2) and (3) are used to calculate species concentration from a single pair of pulses.

The advantage of heterodyne systems in opposition to direct-detection DIAL systems comes about because of the higher SNR—defined as the average signal power in Eq. (1) divided by the noise power—at the output of the detector. Direct-detection systems are more desirable when high SNR can be attained. Coherent DIAL is interesting as part of a versatile system, which combines the measurement of wind and molecular species. However, power measurement errors, which result from fluctuations in the instantaneously received signal caused by speckle and turbulence, have a much stronger effect on coherent detection than on direct-detection DIAL systems.

The accuracy of the estimate of average received power at the two wavelengths is actually the critical parameter in coherent DIAL profiling. Any relative error in the power measurement resulting from atmospheric turbulence will translate as a relative error in the estimation of absolute concentration: Fluctuations in the instantaneous power level, which do not affect the average power, nonetheless degrade the ability of the system to measure this average power. Atmospheric refractive turbulence produces signal fluctuations, which affects heterodyne detection systems in different ways and, therefore, it must be a consideration in the proper evaluation of DIAL-system reliability. Although power fluctuations could result from a number of physical mechanisms other than refractive turbulence (i.e., aerosol variability, speckle, detection noise, and background noise), these mechanisms are not the focus of this analysis and will not be discussed here. In addition, since we are mainly interested in refractive turbulence effects, in Eq. (4) we assume that the absorption coefficient K uncertainty—caused by the inexact knowledge in line parameters attached with uncertainties in temperature and pressure within the measurement volume—is negligible. All these very relevant considerations have been thought over carefully elsewhere (see, for example, Refs. [11–13]). Our results regarding turbulence effects complement those earlier analyses. Finally, it is necessary to mention that all the results shown in this study will describe single-shot uncertainties. For N independent shots the standard deviation of our measurements would be reduced by 1/√N. As before, being concerned with turbulence-induced errors, in this study we will not consider any temporal averaging of the signal fluctuations.

To undertake the estimation of relative error in DIAL concentration measurements, we can relate the normalized variance of the measured quantity to the normalized variance of the power measurement that results from turbulence index fluctuations. The calculated error (normalized standard deviation) is based on linearized error analysis, which assumes small fluctuations of the variables compared with their mean values [17]. In Eq. (2), the propagation of errors associated to the measurement of ρ(R) that depends on the four values of received signal power at the two wavelengths and two range gate positions leads to

where ρ̄ is the true molecular concentration and

Here ${{\mathrm{\sigma }}_{\mathrm{\rho }}}^2$, ${{\mathrm{\sigma }}_{\mathrm{\delta }\mathrm{P}}}^2$, and ${{\mathrm{\sigma }}_{\mathrm{P}}}^2$ are the normalized variances of the subscripted parameters. We have used the fact that the variance on the logarithm of the variable δP is just its normalized variance. Examining Eqs. (4) and (5) we see that uncertainty in DIAL concentration ${{\mathrm{\sigma }}_{\mathrm{\rho }}}^2$ is expressed through the sample variance of the power fluctuations. Turbulence-induced power fluctuations at different ranges could be correlated: When fluctuations in the power terms in Eq. (3) are not independent, covariance among them becomes nonzero, and an improvement in the overall measurement accuracy will result. The covariance terms are included in Eq. (5) through the term O(R):

Here, C_P (R-ΔR/2, R+ΔR/2) represents the normalized covariance between the fluctuations of the measured received power P at ranges R-ΔR/2 and R+ΔR/2. The final term O² (R) in Eq. (6) covers any other possible correlation terms, such as the correlation of turbulence-induced power fluctuations at different wavelengths. As a first approach to the problem, in most practical situations we can assume that O² (R) is sufficiently small to be overlooked.

The same Eqs. (4) and (5), used to describe DIAL measurement uncertainty in the presence of turbulence, can express speckle effects. In this case, the variance of the power fading ${{\mathrm{\sigma }}_{\mathrm{P}}}^2$ is unity for any wavelength and range considered in the problem. Also, fluctuations in received power due to speckle are uncorrelated at different ranges, and the covariance term O(R) depicted in Eq. (6) vanishes. Therefore, the magnitude ${{\mathrm{\sigma }}_{\mathrm{\delta }\mathrm{P}}}^2$,, portrayed in Eq. (5) and responsible for the speckle uncertainty in DIAL concentration is always equal to 4 (i.e., an equivalent standard deviation of 3 dB around the mean power values).

In contrast, as shown in Fig. 1, this power equivalent variance depends strongly on range when turbulence fluctuations are considered. It is necessary that the error analysis estimate the normalized power variance ${{\mathrm{\sigma }}_{\mathrm{P}}}^2$ and covariance C_P [10]. Figure 1 shows the variance ${{\mathrm{\sigma }}_{\mathrm{\delta }\mathrm{P}}}^2$ of the power-ratio δP, in a coherent DIAL system, as a function of range R and moderate-to-strong refractive turbulence ${{\mathrm{C}}_{\mathrm{n}}}^2$ for a 2-µm wavelength, 16-cm aperture, monostatic lidar system. The power variance, calculated with our simulation technique, is shown for different range resolutions ΔR. Both variance and covariance terms in Eq. (5) are considered in our simulations. (In Ref. [10], using the target-plane formulation, we showed first how the statistical properties of power P are those that correspond to the overlap integral of the transmitted and virtual backpropagated local oscillator irradiances at the target plane; and, second, we described the estimation process of both variance and covariance of power fluctuations.) All simulations assume uniform turbulence with range and use the Hill turbulence spectrum [18] with typical inner scale l₀ of 1 cm and realistic outer scale L₀ of the order of 5 m. Also, we surmise no range dependency in the scattering coefficients β and α. The simulation technique uses a numerical grid of 1024×1024 points with 5-mm resolution and simulates a continuous random medium with a minimum of 20 two-dimensional phase screens [7]. We run more than 4000 samples to reduce the statistical uncertainties of our estimations to less than 2% of their corresponding mean values.

In any case considered in Fig. 1, the power equivalent variance term ${{\mathrm{\sigma }}_{\mathrm{\delta }\mathrm{P}}}^2$ is below 0.25, i.e., an equivalent standard deviation of -3 dB around the mean power values. This makes the turbulence-induced equivalent standard deviation at least four times (6 dB) smaller than that of speckle fading. Still, it is significant, at most levels of turbulence in ground-coherent DIAL systems now in use or under development that work at wavelengths that range from 1 to 10 µm [19].

3. DIAL accuracy degradation due to atmospheric turbulence

Inescapable errors in coherent power measurement give rise to inescapable errors in DIAL results. Applying Eq. (4) to the results for ${{\mathrm{\sigma }}_{\mathrm{\delta }\mathrm{P}}}^2$, shown in Fig. 1, provides a straightforward estimate of the variance ${{\mathrm{\sigma }}_{\mathrm{\rho }}}^2$ in DIAL concentration measurements for any molecular species of interest. Also, the linear formulation of the DIAL concentration uncertainty helps, inestimably, to physically interpret the outcomes of our raw simulations. However, it is advisable to proceed with caution and take notes. The lineal analysis drawn by Eq. (4) deals with the way that power errors combine and propagate to give errors in concentration measurements. In any case, these are approximate laws, valid only for small errors, meaning that the fractional error considered must be much less than unity. Although the laws allow us to estimate the error in DIAL measurement with some degree of accuracy, even as the small fluctuations approach is not fully substantiated, we may certainly be assuming an unspecified amount of ambiguity in our assessment.

 

Fig. 1. Coherent power equivalent variance in a DIAL system as a function of range R and different moderate-to-strong refractive turbulence ${{\mathrm{C}}_{\mathrm{n}}}^2$ daytime values for a 2-µm wavelength, 16-cm aperture, monostatic lidar system. The power variances are shown for different range resolutions ΔR. The dashed curve and y-axis labeling on the right corresponds to the mean coherent power.

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Conveniently, simulation permits us to overcome this difficulty with the use of another approach to the estimation of relative error in DIAL-concentration measurement that bypasses the limitations of Eq. (4). It relies on the use of Eq. (2) to compute the instantaneous, simulated concentration ρ(R) for a specific set of atmospheric parameters [until now we have used our simulations to estimate instantaneous coherent received power, as described in Eq. (1)]. After running sufficient realizations to reduce estimated statistical uncertainties, we can assess the total standard deviation σ_ρ of the mean concentration measurement resulting from refractive turbulence. The molecular concentration values at specific ranges for each realization are combined to calculate the error. This procedure does not need to assume small concentration turbulence-induced fluctuations and, consequently, it is more flexible than the linearized DIAL error analysis given by Eq. (4). In using this path, of course, we will also consider independent power fluctuations at different wavelengths and deterministic absorption coefficients.

Figures 2–3 show the normalized standard deviation (relative error) of DIAL measurement fluctuations as a function of range and different levels of refractive turbulence ${{\mathrm{C}}_{\mathrm{n}}}^2$ for 2-µm and 10-µm monostatic systems with 16-cm apertures (other simulation parameters are similar to those described previously for Fig. 1). As bistatic lidar configuration tends to be just slightly more immune to turbulence-induced power fluctuations than monostatic arrangements [10], most of the remarks of this study about DIAL uncertainty in monostatic systems will also apply for bistatic situations. Therefore, we will synthesize and will not report bistatic results here. Several sounding frequency candidates that are capable of discerning atmospheric gaseous species concentrations have been identified for DIAL implementations. Table 1 exhibits the wavelength selection of laser lines for range-resolved remote sensing of atmospheric constituents H₂O, CO₂, and NH₃ by use of the coherent DIAL systems that are considered in this study. We compute the instantaneous, simulated concentration ρ(R) as portrayed by Eq. (2) for the set of line coefficients depicted in Table 1. Line parameters presume a standard atmospheric pressure of 760 Torr and a reference temperature of 296 K.

In Fig. 2, the relative error in a 2-µm DIAL concentration measurement is shown under typical diurnal conditions of strong-to-moderate turbulence. The normalized standard deviations are shown for different DIAL range resolutions ΔR. Range-resolved DIAL

Table 1. Wavelength selection of laser lines for range-resolved coherent DIAL measurements.

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measurement errors of CO₂ concentration are compared with the relative errors that correspond to measurements of H₂O concentration. The slight differences between the two concentration measurement uncertainties are due to the particularities of the chosen line parameters and the concentration levels assumed in the simulations (see Table 1). Roughly, any other possible wavelength pair or species concentration will modify the products of our simulations display in the figures, according to Eq. (4). For any range considered in the figure, the concentration measurement relative error is usually below unity (100 percentage error). When the shortest ranges and strong turbulence conditions are studied, the measurement error can approach a maximum of 1.5 (see Fig. 2, left). Certainly, with ground lidar systems profiling the atmosphere along slant paths with large elevation angles, the accumulated turbulence level and its effects should be noticeably smaller. In many practical situations, however, when the atmospheric measurement of interest is concentration versus height (atmospheric profiles), elevation angles are usually small enough to maximize the number of independent samples for a given altitude resolution.

 

Fig. 2. Turbulence-induced DIAL measurement uncertainty as a function of range R and different moderate-to-strong refractive turbulence ${{\mathrm{C}}_{\mathrm{n}}}^2$ daytime values for a 2-µm wavelength, 16-cm aperture, monostatic lidar system. The standard deviations are shown for different DIAL range resolutions ΔR. Range-resolved DIAL measurement errors of CO2 (carbon dioxide) concentration (solid curves) are compared with those relative errors corresponding to measurements of H₂O (water vapor) concentration (dashed curves) (see text for further details).

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The lineal analysis presented in Section 2, which shows the main variance and covariance terms affecting the concentration measurement uncertainty, will, along with the concept of target plane beam averaging [10], allow us to comprehend the DIAL behavior displayed in the figures. In general, we should expect that a large number of bright spot scintillations in the target plane would increase beam averaging and, consequently, reduce power fluctuations, i.e., DIAL uncertainty. The maximum of the concentration error in Fig. 2 at shorter ranges is mostly due to the fact that a single scintillation fills the beam area in the scattering target. With different strength, a similar maximum can be appreciated for all range resolutions ΔR that are considered.

For sharper lidar range resolutions, as the thickness of the medium ΔR decreases, the physical significance of the correlation terms describing the power fluctuations becomes more important (power fluctuations are correlated along the lidar profiling path because of the propagation of beam turbulence effects from the plane R to the plane R+ΔR). Consequently, the improvement associated with the correlation of the power estimations in the overall DIAL-measurement accuracy becomes larger. On the other hand, the decrease of the range resolution ΔR translates into a smaller spatial averaging of turbulence fluctuations. As a result, in spite of the reduction of the covariance terms, in Fig. 2 the relative error, which is induced by turbulence in the measurement process, decreases with increasing ΔR.

 

Fig. 3. DIAL error resulting from atmospheric turbulence as a function of range R and typical daytime conditions of strong turbulence ${{\mathrm{C}}_{\mathrm{n}}}^2$ for a 10-µm wavelength, 16-cm aperture, monostatic lidar system. The concentration relative errors are shown for different DIAL range resolutions ΔR. Range-resolved DIAL measurement errors of NH₃ (ammonia) concentration (solid curves) are compared with those errors corresponding to measurements of H₂O (water vapor) concentration (dashed curves).

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For growing ranges, the beam resolves several scintillation spots, producing an increase of the averaging effect. Also, at far ranges, beam intensity saturation (scintillation reaches a maximum and then stays there or decreases slowly) limits the continuous increases of power fluctuations which, as a result, will show a high correlation along the lidar path. The effect is most apparent when strong turbulence is considered in Fig. 2 (left), where correlation terms almost completely compensate variance terms, bringing DIAL uncertainty down to near zero. For moderate turbulence levels, shown in Fig. 2 (right), this effect is less pronounced, and an important residual error still remains at far ranges.

In Fig. 3, a somewhat different behavior can be observed for the relative error in a 10-µm DIAL concentration measurement when the same typical diurnal conditions of turbulence that were used in Fig. 2 are considered. Once again, concentration relative errors are shown for different DIAL range resolutions ΔR. Range-resolved DIAL measurement errors of NH₃ concentration are compared with those errors corresponding to measurements of H₂O concentration. The first, and most arresting, observation from these results is the fact that the uncertainty added by turbulence on the DIAL measurement process of a 10-µm lidar may apparently be even higher than the error predicted for a 2-µm system, where we would anticipate a superior sensitivity to turbulence. On the basis of this uncertainty analysis, a 2-µm system seems to outperform an equivalent 10-µm lidar because of the presence of refractive turbulence. Also of interest, as shown in Fig. 3, is the fact that the error shows little dependency with range. Uncertainty seems to behave evenly for most of the ranges considered in the figures. When strong ${{\mathrm{C}}_{\mathrm{n}}}^2$ levels are considered in our simulations (see Fig. 3, left), concentration measurement errors can reach consistently values of 2 for almost any range analyzed. For moderate turbulence (see Fig. 3, right), DIAL uncertainty is predicted to be approximately a factor of 2 smaller.

To explain these results of our simulations, we must consider the fact that beam averaging in the situations depicted in Fig. 3 is now notably smaller than under the conditions drawn in Fig. 2. Paradoxically, the smaller sensitivity to turbulence of 10-µm beams translates into larger intensity scales in the target plane, and prevents the leveling of power fluctuations with beam averaging. As a consequence, DIAL uncertainties are more prevalent in 10-µm systems than in the 2-µm lidars. The effect is predominant under the conditions of high turbulence, as shown in Fig. 3 (left). Although the intensity scales eventually decrease for larger ranges, the consequent increases in beam averaging compensated for the growth of power fluctuations: For higher wavelengths, intensity fluctuations reach saturation levels much more slowly, so power fluctuations may increase continuously for all the ranges considered in the figures. The rise of scintillation makes up for the improvement in beam averaging and thus results in the nearly non-range-dependent DIAL measurement uncertainty observed in Fig. 3.

4. Conclusions

This novel study of the uncertainty added to the coherent DIAL-measurement process by refractive turbulence fluctuations was made possible by the use of our simulation technique. The approach allows us to grasp phenomena beyond the range of available analytical theory. The results indicate a rather complicated behavior, which is strongly affected by atmospheric turbulence conditions and lidar configurations (see Section 3). When 2-µm systems are considered, DIAL uncertainty shows a characteristic maximum at shorter ranges to decrease monotonically afterward. A different mannerism can be observed for the 10-µm lidar, where DIAL percentage error now remains almost flat for any considered range. Regarding the refractive turbulence uncertainty analysis, a 10-µm system seems, surprisingly, to do worse than an equivalent 2-µm lidar. Once again, the concept of beam averaging in the target plane is a simple way of interpreting these results.

Turbulence-induced power measurement uncertainties are not as large as that of speckle fading (see Section 2). Still this power uncertainty may become a greater problem in coherent DIAL systems, owing to the long time constant associated with fluctuations, the large spatial correlations among power variations, and the lack of alignment between the on-line and the off-line beams.

For a system with a sufficiently high pulse-repetition rate, the speckle-induced fluctuations are usually reduced by temporal averaging. The time scale of the fluctuations caused by turbulence is several orders of magnitude larger than that of speckle-induced fluctuations (milliseconds rather than microseconds) and, consequently, turbulence-induced fluctuations cannot be easily averaged out. We may be forced to increase the averaging time required to reduce the measurement uncertainty, thus limiting the effective bandwidth of the lidar system. Also, because of the correlation of turbulence-induced DIAL measurement fluctuations at different ranges, the covariance terms become nonzero and the number of independent range bins will decrease. Although useful in dealing with speckle, spatial averaging of turbulence fluctuations to improve the overall measurement accuracy would be difficult. Finally, although power fluctuations at two DIAL wavelengths λ _on and λ _off may be mostly correlated (i.e., weak wavelength dependency of turbulence effects on the wavelength region of interest) overall except the smallest refractive index eddy sizes, almost simultaneous shot-pair may neither be the solution to the turbulence problem. In most circumstances the performance of the lidar is strongly degraded by the unavoidable lack of alignment between the on-line and the off-line beams, and misalignment tends to wipe out nearly all the possible benefits associated with the correlation of turbulence-induced DIAL-measurement fluctuations at two DIAL wavelengths.

All the previous considerations may prevent, to a certain extent, the ability to take advantage of any of the techniques regularly used to wipe out speckle effects when dealing with the turbulence uncertainty added to the DIAL measurement process of any practical lidar. Accordingly, refractive turbulence may become as prevalent as speckle in limiting DIAL accuracy. We have refrained from considering these important points in detail here. We mean to address them in a subsequent study.

This research was partially supported by the Spanish Department of Science and Technology MCYT grant REN 2003-09753-C02-02.